By Nicholas Vinocur and Emile Picy

PARIS (Reuters) - France's lower house of parliament voted on Tuesday to pass a draft law overhauling the labor system despite widespread union protests against the bill, opening the way for a debate in the Senate later this month.

The vote came as Socialist President Francois Hollande spent the last week fending off accusations of a cover-up after his former budget minister admitted lying about a secret foreign bank account at a time when the government is already struggling to reverse a rise in unemployment and meet its economic targets.

Accusations among some on the left that Hollande has abandoned Socialism could drag down his already dismal approval ratings, stuck around 27 percent, and hurt his party's performance in municipal elections next year.

The National Assembly adopted the draft bill, which ushers in a series of measures to loosen firing and hiring rules, by 250 votes in favor versus 26 against as scores of opposition conservatives abstained and thousands protested outside.

Socialist lawmakers and center-left allies backed the bill but the Greens Party, part of Hollande's government and supposed to be a parliament ally, abstained in a rebuke to the man they helped bring to power last year.

"This is a small revolution," said Annick Lepetit, head of the Socialist group in the National Assembly.

Thirty-five Socialist deputies joined Greens dissidents in abstaining from the vote, hinting at a growing rift in Hollande's camp as tens of thousands of protesters led by the hardline CGT union marched against it in 170 towns and cities.

In all, 178 lawmakers abstained, mainly conservatives.

Street protests will not block the reform from becoming law because Hollande, whose Socialist Party has a majority in parliament, is likely to find enough support in the Senate to pass the law when the debate moves there later in April.

But hardline unions, aided by their left-wing allies in parliament, aim to mobilize public opinion against the bill and obtain as many amendments as possible during the debate.

Members of the CGT, France's second-largest union by membership, branded the draft law as "traitorous", splitting with its bigger rival, the CFDT, which backed the labor overhaul in an accord reached after weeks of talks.

In the southern city of Marseille, marchers bore banners reading: "No to breaking the labor code", against a text seen as the most comprehensive labor reform since World War Two.

"We won't let anything past us," regional CGT chief Mireille Chessa told reporters.

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Public opinion on the reforms is divided. But a survey by pollster BVA in March found 62 percent of respondents supported the bill, making it more popular than Hollande himself.

"The point is ... to make workers aware of the impact this is going to have on their daily lives," Thierry Lepaon, head of the CGT union, told Canal+ television.

In dozens of articles, the bill aims to inject more flexibility into a system the IMF, OECD and ratings agencies say is to blame for chronically high unemployment because it gives some workers too much job security and others too little.

Pressure from Brussels to fix structural brakes on competitiveness, from rigid labor markets to tax fraud, compelled leaders in Italy and Spain to undertake similar profound reforms despite opposition on the streets.

Labor Minister Michel Sapin has said the changes will be implemented as soon as possible in a bid to combat unemployment, which has risen steadily over the past two years to reach 10.6 percent, close to an all-time record in jobless claims.

But far-left lawmakers, who have sought minor amendments to weaken the bill, hope to slow its passage through the Senate.

"We are acting as amplifiers for the demands of the protesters today," Andre Chassaigne, head of the Communist Party's group in parliament, told left-wing paper l'Humanite.

Laurence Parisot, head of the Medef employers' union, said she was happy overall with the draft bill.

"But careful," she added, on BFM TV. "I would not stand for it if the nibbling that went on in the Assembly were to become bigger bites in the Senate."

(Additional reporting by Jean-Francois Rosnoblet in Marseille; Editing by Catherine Bremer and Michael Roddy)